Let $I=[0,1)$. The sawtooth (Bernoulli shift) map $F: I \rightarrow I$ is defined by

$$F(x)=2 x[\bmod 1]$$

Describe the effect of $F$ using binary notation. Show that $F$ is continuous on $I$ except at $x=\frac{1}{2}$. Show also that $F$ has $N$-periodic points for all $N \geqslant 2$. Are they stable?

Explain why $F$ is chaotic, using Glendinning's definition.